In this thesis we study modifications of the classical Mean-Variance Portfolio Optimization model. Our objective is to identify an optimal subset of assets from all available assets to maximize the expected return while incurring the minimum risk. In addition, we test several approaches to measuring the effect of the variance of the portfolio on the optimal asset allocation. We have developed a mixed integer formulation to solve the well known Markowitz portfolio model. Our model captures and solves the certain practical drawbacks that a real investor would face with the Markowitz approach. For example, by selecting a limited number of assets our procedure tends to prevent small allocations of assets. In addition, we find that in most cases, the maximum drawdown increases as a function of the upper bound on the variance of the portfolio and that this result is consistent with intuition, since portfolio risk increases as the chance that a drawdown event occurs also increases. However, we have observed that altering the composition of the portfolio can mitigate the risk of a drawdown event.